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TSpace Research Repository tspace.library.utoronto.ca
Electro-optic Response in Germanium Halide
Perovskites
Grant Walters and Edward H. Sargent
Version Post-print/accepted manuscript
Citation (published version)
Walter, G.; Sargent E.H. Electro-optic Response in Germanium Halide Perovskites. J. Phys. Chem. Lett., 2018, 9, pp 1018–1027.
Publisher’s Statement
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1
Electro-Optic Response in Germanium Halide
Perovskites
Grant Walters, Edward H. Sargent*
Department of Electrical and Computer Engineering, University of Toronto, 35 St. George
Street, Toronto, Ontario, M5S 1A4, Canada
Corresponding Author
*Email: [email protected]
2
ABSTRACT Electro-optic materials that can be solution-processed and provide high crystalline
quality are sought for the development of compact, efficient optical modulators. Here we present
density functional theory investigations of the linear electro-optic coefficients of candidate
materials cesium and methylammonium germanium halide perovskites. As with their lead halide
counterparts, these compounds can be solution-processed; but in contrast, they possess the non-
centrosymmetric crystal structures needed to provide a linear electro-optic effect. We find
substantial electro-optic responses from these compounds; in particular, for the r51 tensor element
of CsGeI3 we predict an electro-optic coefficient of 47 pm·V-1 at the communications wavelength
of 1550 nm, surpassing the strongest coefficient of LiNbO3 at 31 pm·V-1. The strong electro-optic
responses of the germanium compounds are driven by high nonlinear susceptibilities and dynamics
of the germanium atoms that ultimately arise from the distorted crystal structures. Alongside the
electro-optic coefficient calculations, we provide the frequency responses for the linear and
nonlinear electronic susceptibilities.
TOC GRAPHICS
KEYWORDS density functional theory, nonlinear susceptibility, Pockels effect, electro-optic
coefficient, dielectric constant
3
The development of efficient, compact electro-optic modulators for use in intra-chip or inter-chip
optical interconnects will be further advanced by expanding the set of electro-optic materials
options. Established inorganic crystals, chiefly LiNbO3 and BaTiO3, are fabricated using
processing techniques, such as Czochralski crystal growth or epitaxial MOCVD, that limit
compatibility with standard silicon photonics. Organic materials are deposited via inexpensive
and simple solution-processing methods and have performed impressively when incorporated
within pioneering modulator architectures such as silicon slot-waveguide interferometers.1,2
Unfortunately, the electrostatic poling of the organic molecules that is needed to orient and
maintain orientation of the molecules has limited their breadth of application to date.
These limitations motivate the search for new solution-processed materials that exhibit a
strong, built-in, electro-optic activity.
Recently, metal halide perovskites have risen as solution-processed materials with
impressive electrical and optical properties. These materials, possessing the chemical formula
ABX3 (where A is a cation, B is a metal cation, and X is a halide anion), have demonstrated
impressive performance in a broad range of optoelectronic devices.3–6 Elements of the success of
metal halide perovskites can be traced to their flexibility in composition and morphology. The
optical, electrical, and structural properties can be finely tuned through different combinations of
cations and anions.7,8 Various solution-processing methods can be used to grow different forms
of perovskites: quantum-confined nanostructures,9–11 polycrystalline thin films,12 and
macroscopic single crystals.13,14
The nonlinear optical properties are much less explored than their behavior as light-
absorbing and light-emitting materials. The widely researched metal halide perovskites,
4
particularly the lead-based ones, are structurally globally centrosymmetric and so are incapable
of some nonlinear optical processes, including the linear electro-optic effect (LEO).
The germanium halide perovskites are a class of non-centrosymmetric compounds. As in
the case of lead, germanium is a group IV element that is capable of carrying a 2+ valence state.
However, its higher position within the periodic column lends the 4s electron pair greater
stereochemical activity when compared to the lead analogue.15 This activity distorts the
perovskite unit cell, causing it to lose its inversion symmetry. The germanium halide perovskites
are drawing increased interest as lead-free alternatives for photovoltaic application,16–19 and have
also previously been reported to exhibit impressive second-harmonic generation (SHG)20–24, a
phenomenon intimately linked with LEO. Motivated by this fact – combined with their
transparency over the infrared communications wavelengths20,25,26, and evidence for growth of
crystals from solution20,22,23,25,27,28 – we investigate herein the electro-optic and nonlinear optical
behavior for the germanium halide perovskites using density functional theory calculations.
We focus our study on the cesium germanium halides (CsGeX3; X = I, Br, Cl) and
methylammonium germanium iodide (MAGeI3). We provide predictions for the LEO
coefficients and, by examining the factors contributing to the electro-optic behavior and the
trends associated with the halide anions and A cations, provide mechanistic insights into the
LEO responses. We complement these predictions with calculations of the linear and nonlinear
susceptibilities and their frequency responses.
As in other metal halide perovskites, germanium-based perovskites assemble as an
inorganic network of metal halide octahedra surrounding their A-site cations, such as
methylammonium or cesium cations. However, in contrast with most other perovskites, those
5
based on germanium have a distorted unit cell: they reside in the non-centrosymmetric trigonal
R3m space group20,22,23,27,29. The rhombohedral representation of the crystal structure is provided
in Figure 1a-c. The angular lattice parameter of the unit cell deviates only slightly from 90
degrees. The A cations occupy the corners, the halide species occupy positions near the face
centers and the germanium atoms occupy positions offset along the [111] direction from the
body centers. For compounds with the polar molecular methylammonium, which possesses C3v
symmetry, the molecule’s C-N axis aligns along the [111] direction20. The atomic offsets yield
two distinct Ge-X bonds. Notably, the central offset of the germanium cations is reminiscent of
that for titanium atoms in the electro-optic material BaTiO3.
Calculations of the structural parameters (Figure 1d-f and Supplementary Table 1) agree
with the experimental findings for synthesized microcrystals20,22,23,27,29. Unsurprisingly, the
cesium compounds show that as the size of the halide anion increases, the unit cell expands. The
unit cell’s angular deviation grows as the cell expands. MAGeI3 has a similar lattice constant as
its cesium counterpart, but it exhibits a much greater angular distortion. The germanium atoms’
offset from the body-center increases as the halide size increases. Accompanying this is a change
in the offset of the halide species from the face-centers that lessens the discrepancy between the
two lengths of Ge-X bonds.
6
Figure 1. Structural representations of the
rhombohedral lattice for germanium halide
perovskites: (a) standard orientation, (b) [100] view, (c)
[111] view showing hexagonal symmetry. Red, black,
and white spheres are Ge2+ cations, A+ cations, and X-
halide anions respectively. Calculated structural
parameters: (d) lattice constant, (e) lattice angular
parameter, (f) XYZ coordinate of the near body-center
Ge atoms. Experimental results are taken from
references 20 and 22.
7
Electro-optic materials, to be of use in optical communications systems, are required to
be optically transparent across the infrared communications bands. As in the case of most metal
halide perovskites, the electronic bandgaps of the germanium compounds decrease as the halide
size increases and lie within the visible spectrum. Substitution of cesium cations with
methylammonium leads to widening of the bandgap, which has previously been attributed to
further stereochemical activition of the germanium 4s2 lone pair20.
We calculated the bandgaps at the level of the LDA, GGA, and HSE06 approximations
(Table 1). We further calculated the real and imaginary components of the dielectric function,
𝜀(𝜔), at optical frequencies (atomic positions are clamped) at the LDA level using scissors
corrections set to the experimentally reported bandgaps; the response curves are provided in
Figure 2. The use of scissors corrections is necessary due to the problem of bandgap
underestimation in DFT and the consequent overestimation of the dielectric properties21,30,31. The
corrections act as rigid shifts to the conduction bands in order to curtail bandgap estimation and
have been widely used in calculations of linear and nonlinear optical properties of many
materials.21,30–46 As empirical adjustments, these corrections introduce a degree of uncertainty to
the calculations; however, as we show below, these corrections are validated by the strong
agreement between the scissors corrected and experimental SHG characteristics.
The germanium materials possess uniaxial birefringence and refractive indices that scale
with bandgap. Along with the dielectric functions we have plotted the electronic band structures
in Figure 3. All of the germanium compounds exhibit a direct bandgap at the Brillouin zone Z
point; the rhombohedral nature of these crystals changes the gap from the R point of cubic
perovskite phases47. The first few optical transitions can be connected with spectral features in
the imaginary dielectric functions.
8
Table 1. Calculated bandgap values (eV).
compound LDA GGA HSE06 Exp.
CsGeCl3 2.01 2.21 2.22 3.43*
CsGeBr3 1.26 1.53 1.66 2.38*
CsGeI3 0.66 1.19 1.41 1.6**
MAGeI3 0.37 1.52 1.84 1.9**
*Reference 22
**Reference 20
Figure 2. Calculated dielectric functions for germanium halide perovskites. (a) Real ordinary
components. (b) Real extraordinary components. (c) Imaginary ordinary components. (d) Imaginary
extraordinary components. Lettering on curves for CsGeI3 is used to connect with electronic band
dispersion plot. Calculations have been done with LDA exchange-correlation functionals and have been
scissors corrected.
9
Figure 3. Calculated electronic band diagrams for germanium halide perovskites: (a) CsGeI3, (b)
CsGeBr3, (c) CsGeCl3, (d) MAGeI3. The first few
optical transitions have been indicated for CsGeI3.
Scissors corrections have been applied and LDA
functionals were used.
10
The LEO effect is the special case of second-order nonlinear optical processes where one
of the interacting electric fields is low-frequency while the other remains at optical frequencies.
Changes to the optical dielectric constants can be induced by the low-frequency electric fields
and are related through LEO coefficients. The LEO effect on the optical dielectric constant is
defined by the following relation,
∆(𝜀−1)𝑖𝑗 = ∑ 𝑟𝑖𝑗𝛾𝐸𝛾3𝛾=1 (1)
where 𝑟𝑖𝑗𝛾 are the LEO coefficients and 𝐸𝛾 are electric field components (Greek indices
correspond to static directional fields, Latin indices correspond to optical directional fields). The
LEO coefficients are the central figures of merit for electro-optic materials; a large LEO
coefficient is key for efficient electro-optic modulation. The LEO coefficients of a material can
be calculated with density functional perturbation theory (DFPT), as formulated by Veithen et
al30. By determining the energetic changes induced by atomic displacements and homogeneous
electric fields using DFT calculations, the electro-optic coefficients can be found.
The LEO coefficients at telecommunications bandwidths (i.e., field frequencies above
100 MHz) are formed from two contributions: an electronic, 𝑟𝑖𝑗𝛾𝑒𝑙 , and an ionic, 𝑟𝑖𝑗𝛾
𝑖𝑜𝑛, response30.
Within the Born-Oppenheimer approximation, these two quantities sum to form the total LEO
coefficient. The electronic contribution is due to field interactions with the valence electrons
while considering the ions as clamped. This term is related to the LEO second-order
susceptibility, 𝜒𝑖𝑗𝑘(2)(−𝜔; 𝜔, 0), via,
𝑟𝑖𝑗𝛾𝑒𝑙 =
8𝜋
𝑛𝑖2𝑛𝑗
2 𝜒𝑖𝑗𝑘(2)(−𝜔; 𝜔, 0)|
𝑘=𝛾 (2)
11
where 𝑛 are refractive indices.30 We note that the DFT calculations of this term neglect the
dispersion of the second-order susceptibility, and so the term typically presents a lower bound.
The ionic contribution accounts for the relaxation of the atomic positions due to the electric field
and the corresponding dielectric changes. This term is given by a sum over the transverse optic
phonon modes (indexed by 𝑚),
𝑟𝑖𝑗𝛾𝑖𝑜𝑛 = −
4𝜋
√𝛺0𝑛𝑖2𝑛𝑗
2 ∑𝛼𝑖𝑗
𝑚𝑝𝑚,𝛾
𝜔𝑚2𝑚 (3)
where 𝛺0 is the unit cell volume, 𝛼𝑖𝑗𝑚 are the Raman susceptibility components for the modes,
𝑝𝑚,𝛾 are the mode polarities, and 𝜔𝑚 are the mode frequencies.30 The Raman susceptibilities are
found from a sum over the products of the changes in susceptibility resulting from atomic
displacements, 𝜕𝜒𝑖𝑗(1)
/𝜕𝜏𝜅,𝛽 , and the modal atomic eigendisplacements, um(κβ), for all atoms
(indexed by 𝜅)30:
𝛼𝑖𝑗𝑚 = √𝛺0 ∑
𝜕𝜒𝑖𝑗(1)
𝜕𝜏𝜅,𝛽𝜅,𝛽 um(κβ) . (4)
The mode polarities are given by a similar sum over the products of the Born effective charges,
Zκ,γβ∗ , and the modal atomic eigendisplacements30:
pm,γ = ∑ Zκ,γβ∗
κ,β um(κβ) . (5)
Two quantities central to the DFPT LEO calculations are the linear and second-order
nonlinear susceptibilities. Table 2 gives the calculated optical (𝜀∞, atomic positions clamped)
and static (𝜀0, atomic positions unclamped) zero-frequency dielectric constants at the LDA level
with and without scissors corrections. The dielectric constants reflect the same conclusions
12
drawn from the calculated dielectric functions previously shown. As expected, the scissors
corrections also reduce the dielectric constants and so will play a role in the determination of the
LEO coefficients. Table 3 provides the LEO nonlinear susceptibilities, given as 𝑑𝑖𝑗 = 𝜒𝑖𝑗(2)
/2
using contracted indices, with and without scissors corrections. Again, the bandgap correction
generates a considerable difference. Akin to the linear susceptibilities, the nonlinear
susceptibilities increase as the size of the halide anion increases. Substitution of cesium with
methylammonium leads to a decrease in the nonlinear susceptibilities. Thus, inspection of all the
compounds shows simply the expected scaling of the nonlinear susceptibility with bandgap48.
Interestingly, the greatest nonlinear susceptibility component for CsGeCl3 and CsGeBr3 is the
𝑑33, while for CsGeI3 and MAGeI3 it is the 𝑑13.
Table 2. Optical and static linear dielectric
properties obtained from 2n+1 theorem DFPT
calculations. compound 𝜀11,22
0 𝜀330 𝜀11,22
∞ 𝜀33∞
CsGeCl3 LDA 9.87 9.05 3.85 3.69 SCI 9.32 8.54 3.30 3.18
CsGeBr3 LDA 12.00 11.30 5.07 4.96
SCI 11.28 10.56 4.34 4.22
CsGeI3 LDA 14.97 14.87 7.48 7.69
SCI 13.53 13.13 6.04 5.95
MAGeI3 LDA 15.26 7.54 5.36 5.42
SCI 14.53 6.73 4.63 4.61
13
Table 3. LEO nonlinear susceptibilities (pm·V-1)
obtained from 2n+1 theorem DFPT calculations.
compound 𝑑11 𝑑13 𝑑33 CsGeCl3 LDA 1.0 4.6 9.5
SCI 0.1 2.0 4.8
CsGeBr3 LDA 9.9 16.7 21.6
SCI 4.1 8.5 12.9
CsGeI3 LDA 104.3 173.4 -32.9
SCI 33.9 50.6 12.1
MAGeI3 LDA 32.0 44.9 8.9
SCI 13.6 20.1 10.3
These calculations of the electronic susceptibilities, along with the ionic terms calculated
by atomic perturbation calculations, were then used to obtain the LEO coefficients. The space
group for the germanium compounds yields eight non-zero tensor components of which four are
unique (Supplementary Figure 1); these happen to be the same components possessed by
LiNbO3. The coefficients for each germanium compound are provided in Table 4. The
contributions from each set of ionic transverse optical (TO) phonon modes and from the
electronic responses are listed at the LDA level. We have further calculated the final coefficients
at two different levels of scissors corrections. The first, SCI1, uses the scissors corrected
dielectric constants. The second, SCI2, uses scissors corrected values for both the dielectric
constants and the nonlinear susceptibilities. We note that this means we have only used scissors
corrections for quantities calculated from electric field perturbations. The differences between
the quantities obtained with the different levels of correction illustrate the uncertainty of the
calculations; however, considering the available reports of the experimental effective SHG
nonlinear susceptibilities for these materials at optical frequencies20,22,23, we believe that the
more modest values of the LEO susceptibilities obtained with the scissors corrections are more
14
accurate and so SCI2 stands as the most accurate prediction of the LEO response. We provide a
plot to illustrate trends in the electronic and ionic responses, with this level of correction, in
Figure 4. All of the germanium compounds exhibit significant LEO responses; CsGeI3 possesses
a component (𝑟51 = 37.64 𝑝𝑚 ∙ 𝑉−1) that even exceeds the strongest component for LiNbO3
(experimental 𝑟33 = 30.8 𝑝𝑚 ∙ 𝑉−1 at 633 nm49; SCI2 calculation 27.28 𝑝𝑚 ∙ 𝑉−1 – see
Supplementary Table S2). This is particularly remarkable considering that these calculated
values exclude any frequency dependence and so represent a lower bound. For the cesium
compounds, the LEO response tends to increase as the halide size increases; this is primarily
driven by the corresponding increases to the electronic contribution. Smaller differences are
observed in the ionic contributions. These differences are most noticeable between CsGeI3 and
MAGeI3, where the methylammonium compound has a considerably smaller ionic contribution
and so a much weaker LEO response.
Further insight into the ionic contributions can be gained by inspecting the modal parts
and their characteristics. First we consider the Born effective charges (Supplementary Table 3).
The effective charges are defined as the changes in polarization that result from atomic
displacements and are factors that control the Coulombic interactions between nuclei50,51. They
are indicative of the influence of dynamical changes to orbital hybridization caused by atomic
displacements50,51. For the germanium perovskites, as the size of the halide anion increases, the
effective charges deviate more from their nominal charges (A+, Ge2+, X-), especially for the Ge
and X atoms, indicating that the bonds become more covalent and sensitive to atomic
displacements. The vibrational modes involving more covalent bonds would then have a greater
electro-optic response. MAGeI3 has bonds with more ionic character than all of the cesium
compounds and the methylammonium cation is almost completely ionized.
15
Table 4. Clamped linear EO coefficients (pm·V-1) obtained from 2n+1 theorem DFPT
calculations. Mode phonon frequencies of ionic contributions are expressed in cm-1.
𝐸 modes 𝐴1 modes compound contribution 𝜔 𝑟11 𝑟51 𝜔 𝑟13 𝑟33 CsGeCl3 TO1 46 0.75 0.42 43 0.45 -0.64
TO2 76 -0.08 0.20 66 -- -- TO3 121 0.20 0.18 148 0.95 1.08 TO4 213 -6.83 -10.53 253 -5.06 -5.62 ionic -5.96 -9.73 -3.66 -5.18 electronic -0.27 -1.29 -1.24 -2.80 total -6.23 -11.02 -4.90 -7.98 total (SCI1) -8.46 -14.90 -6.65 -10.75
total (SCI2) -8.14 -13.92 -5.71 -8.88
CsGeBr3 TO1 43 -0.21 -0.78 35 0.10 -0.79 TO2 53 0.04 0.10 49 -- -- TO3 79 0.22 0.36 93 0.17 0.25 TO4 145 -11.47 -18.07 174 -7.72 -8.55 ionic -11.42 -18.39 -7.45 -9.09 electronic -1.54 -2.65 -2.60 -3.43 total -12.96 -21.04 -10.04 -12.52
total (SCI1) -17.68 -28.89 -13.70 -17.29 total (SCI2) -16.45 -27.12 -11.98 -15.45
CsGeI3 TO1 35 -0.62 -0.88 27 0.28 0.16 TO2 42 0.00 0.00 39 -- -- TO3 60 0.61 0.80 70 -0.67 -0.50 TO4 126 -12.43 -19.94 147 -6.04 -5.67 ionic -12.44 -20.02 -6.44 -6.01 electronic -7.45 -12.05 -12.39 2.23 total -19.89 -32.07 -18.83 -3.79
total (SCI1) -30.48 -51.30 -28.85 -6.32 total (SCI2) -22.77 -37.64 -15.40 -11.40
MAGeI3 TO1 18 3.00 0.16 15 -- -- TO2 45 -0.17 -0.26 73 -2.54 -2.25 TO3 57 0.86 0.98 121 1.60 1.26 TO4 92 0.19 1.18 124 -- -- TO5 144 -6.64 -10.99 163 -3.13 -3.16 TO6 869 0.02 0.02 335 -- --
TO7 1215 0.00 0.00 1004 0.00 -0.00 TO8 1431 0.01 0.01 1394 0.00 -0.00 TO9 1561 -0.00 -0.00 1445 0.04 -0.04 TO10 2762 0.01 -0.02 2792 -0.01 -0.05 TO11 2922 -0.00 0.00 2857 0.01 0.00 ionic -2.73 -8.91 -4.03 -4.24 electronic -4.45 -6.18 -6.24 -1.21 total -7.17 -15.09 -10.28 -5.45
total (SCI1) -9.62 -20.53 -13.78 -7.52 total (SCI2) -6.19 -15.89 -9.16 -7.78
--, not Raman active
16
Next we consider the characteristics of each mode. The vibrational frequencies of the
modes are provided in Supplementary Table 4 and the modal atomic eigendisplacements are
provided in Supplementary Tables 5 to 8. First, we note that the modal frequencies follow an
expected Hookean-type relation to the mass of the halide anions, so that the iodide compounds
benefit the most from the inverse relation of the electro-optic response to modal frequency
squared. We see that the modes involving the covalent germanium halide bonds and
displacements of the germanium atoms possess the greatest electro-optic activity. The
displacement of these atoms, having large effective charges, leads to strong modal oscillator
strengths (Supplementary Table 9) and Raman susceptibilities (Supplementary Tables 10 and
11). These trends then propagate to those observed for the electro-optic activity. Looking at the
cesium compounds, the first set of transverse optical modes (modes 4-6, TO1) involve
movements of the whole germanium halide octahedra, and so exhibit weak LEO responses. The
second set of transverse optical modes (modes 7-9, TO2) consist of an inactive symmetric mode
and a doubly degenerate mode involving small germanium displacements and complex motions
Figure 4. Calculated ionic and electronic
LEO responses with scissors-corrected linear
and nonlinear susceptibilities (SCI2).
17
of the halide anions; the third set of transverse optical modes (modes 10-12, TO3) involve
similar motions as the former’s doubly degenerate mode. Altogether the TO2 and TO3 modes
have weak LEO responses. The fourth set of transverse optical modes (modes 13-15, TO4)
involve large displacements of the germanium atoms and so dominate the ionic LEO response.
MAGeI3 has similar vibrational modes as CsGeI3, but also has high-frequency modes relating to
the internal degrees of freedom of the methylammonium cations. These additional modes have
exceptionally small LEO responses. Furthermore, the strong ionic character of this compound
softens the LEO response of the modes involving germanium and halogen movements.
As a complement to our calculations of the LEO coefficients, we have calculated the
frequency responses of the SHG and LEO second-order electronic susceptibilities
(𝜒𝑖𝑗𝑘(2)(−2𝜔; 𝜔, 𝜔) and 𝜒𝑖𝑗𝑘
(2)(−𝜔; 𝜔, 0) respectively). The below-bandgap absolute response
curves are provided in Figure 5a-d for the LEO and SHG tensor components of the strongest
LEO coefficients and for the effective SHG susceptibilities at the scissors corrected LDA level.
The full frequency responses of the strongest SHG susceptibility tensor components are provided
in Figure 5e. The effective SHG susceptibilities rely on all of the tensor components and
correspond to the values measurable with powder-SHG experiments. Prior reports of the
effective SHG susceptibility at particular wavelengths agree well with our frequency response
calculations: MAGeI3 𝜒𝑒𝑓𝑓(2) (−2𝜔; 𝜔, 𝜔) = 161 pm·V-1 at ~0.7 eV,20 CsGeI3 𝜒𝑒𝑓𝑓
(2) (−2𝜔; 𝜔, 𝜔) =
125 pm·V-1 at ~0.7 eV,20 CsGeBr3 𝜒𝑒𝑓𝑓(2) (−2𝜔; 𝜔, 𝜔) ≈ 18 pm·V-1 at 0.98 eV,22 CsGeCl3
𝜒𝑒𝑓𝑓(2) (−2𝜔; 𝜔, 𝜔) ≈ 2 pm·V-1 at 0.98 eV.22 This agreement also justifies the necessity of the
scissors corrections (see Supplementary Figure S2). The LEO susceptibilities display frequency
dependence that translates to dependence in the electronic contributions of the LEO coefficients
18
(Figure 5f). Accounting for this and the linear susceptibility dispersion, the 𝑟51 LEO coefficient
of CsGeI3 will then increase to 47 pm·V-1 at the telecommunications band of 1550 nm.
In summary, we have determined that the germanium halide perovskites exhibit
significant electro-optic responses that for some compounds are on par with or even exceed that
of the archetypal electro-optic material LiNbO3. The intrinsically distorted nature of the
structures leads to nonlinear electronic susceptibilities and dynamics of the germanium and
halogen atoms that drive the electro-optic activity. The nonlinear susceptibilities and electro-
optic responses are strongest for the iodide compounds and scale with bandgap. The ionic
contributions to the electro-optic characteristics are strongly influenced by the covalency of the
germanium halide bonds, and the vibrational properties of the germanium and halogen atoms.
Substitution of cesium with methylammonium is detrimental to the ionic response, as the bonds
become more ionic and the internal vibrations of methylammonium do not contribute. The strong
electro-optic performance of the germanium halide perovskites, compounded by their solution-
processability, makes them attractive candidates for use in optical modulators.
The realization of germanium halide perovskite optical modulators requires surmounting
of a number of experimental challenges. Crystallization techniques must be developed for these
materials that produce high-quality macroscopic single crystals. Bulk characterization and
implementation in a practical modulator will require an ~ order of magnitude increase in crystal
dimensions over the crystals produced from known techniques, which possess dimensions less
than 100 microns.20,28 As well, the crystals must also be of high optical quality in order to
prevent any influence of internal or surface scattering on the optical characterization. Recent
advances in crystallizing the broader spectrum of metal halide perovskites suggest promising
avenues.14,52–55 Techniques must also be developed for the deposition or growth of the crystals
19
on a modulator platform. The crystals will need to be precisely oriented with the device
architecture and have excellent high optical quality in order to maximize the device efficiency.
Again, recent reports on lead halide perovskites are encouraging in this regard.56–60 The present
study may help to motivate future endeavors to advance the crystallization and deposition
techniques of the germanium halide perovskites with the aim of developing optical modulators.
COMPUTATIONAL METHODS
Calculations of the nonlinear response functions were done within density functional
perturbation theory, employing the 2n+1 theorem, as developed by Veithen et al.30 and
implemented in the ABINIT software package61–66. The package can currently only implement
Figure 5. Calculated frequency response of the electronic nonlinear susceptibility. Below-
bandgap responses for the effective SHG susceptibility, and the SHG and LEO
susceptibilities for the strongest LEO tensor components for (a) CsGeI3, (b) CsGeBr3, (c)
CsGeCl3, and (d) MAGeI3. (e) Full frequency responses of the effective SHG
susceptibility. (f) Normalized frequency response of the strongest electronic LEO terms
for each germanium compound. Calculations were done at the level of the LDA and have
been scissors corrected.
20
nonlinear response function calculations in the Local Density Approximation (LDA) with norm-
conserving pseudopotentials. These calculations include the static and optical dielectric
constants, the nonlinear optical coefficients, the electro-optic coefficients, and the quantities
required in their derivation. Self-consistent calculations were done in the LDA with the
exchange-correlation functional Perdew-Wang 92 parameterization67. Norm-conserving
pseudopotentials generated using the Troullier-Martins method68 were used. Calculations were
done with the experimental lattice constants and with atomic positions relaxed such that maximal
forces were less than 10-6 Ha·bohr-1. Where indicated, a scissors correction has been applied for
all calculations to account for the LDA underestimation of the bandgap and consequent
overestimation of the dielectric properties. The scissors corrections has been included at two
different levels: SCI1, which includes scissors corrected dielectric constants; and, SCI2, which
includes scissors corrected dielectric constants and nonlinear susceptibilities. The Brillouin zone
was sampled using a Monkhorst-Pack 18 × 18 × 18 grid of special k-points, and wave functions
were expanded in plane-waves up to a kinetic energy cut-off of 60 Ha. These parameters were
found to be necessary for convergence of the electro-optic coefficients. The calculations of the
electro-optic coefficients were benchmarked against calculations for LiNbO3, which are provided
in Supplementary Table S2. These results agree well with prior DFT calculations30 and with
experimental findings49.
The frequency dependent optical responses were calculated with the ABINIT61–66
package Optic following work by Hughes and Sipe35, which uses the independent particle
approximation. These calculations include the dielectric function 𝜀(𝜔), the Second-Harmonic
Generation (SHG) susceptibility 𝜒(2)(−2𝜔; 𝜔, 𝜔), and the Linear Electro-Optic (LEO)
susceptibility 𝜒(2)(0; 𝜔, 𝜔). The LEO susceptibility is calculated in the clamped-lattice
21
approximation, which corresponds to the intermediate frequency regime where lattice vibrations
are frozen out and electronic dispersion can be neglected. Self-consistent calculations were done
in the Local Density Approximation (LDA) with the exchange-correlation functional Perdew-
Wang 92 parameterization67. Norm-conserving pseudopotentials generated using the Troullier-
Martins method68 were used. Calculations were done with the experimental lattice constants and
with atomic positions relaxed such that maximal forces were less than 10-6 Ha·bohr-1. A scissors
correction has been applied for all calculations to account for the LDA underestimation of the
bandgap. The Brillouin zone was sampled using a Monkhorst-Pack 26 × 26 × 26 grid of special
k-points, wave functions were expanded in plane-waves up to a kinetic energy cut-off of 20 Ha,
and the number of bands included was 36 for the inorganic compounds and 40 for the
methylammonium compound. These parameters were found to be necessary for convergence of
the nonlinear susceptibilities. The effective nonlinear coefficients were calculated based on
relations provided by Kurtz and Perry69.
Band structure calculations were conducted through ABINIT using the same parameters
as in the preceding paragraph. The Brillouin zone was sampled following a k-path for
rhombohedral structures as described in Reference 70.
Hybrid DFT calculations of the bandgaps were done with the Quantum Espresso71
implementation package. Norm-conserving pseudopotentials with Perdew-Burke-Ernzerhof72
exchange correlations generated with the Martins-Troullier method68 were combined with
HSE06 hybrid functionals73,74 and 0.25 exchange fractions and 0.106 screening parameters were
used. Lattice constants and atomic positions were simultaneously relaxed, such that forces were
less than 2 × 10-6 Ha·bohr-1 and pressures were less than 0.5 kbar, prior to hybrid calculations.
22
An 8 × 8 × 8 Monkhorst-Pack k-point grid and 2 × 2 × 2 q-point grid were used with an energy
cut-off of 30 Ha for the hybrid calculations.
Atomic illustrations were produced with the visualization software VESTA75.
AUTHOR INFORMATION
Notes
The authors declare no competing financial interests.
ACKNOWLEDGMENTS
The work presented in this publication was supported by funding from an award (KUS-11-009-
21) from the King Abdullah University of Science and Technology, the Ontario Research Fund,
the Ontario Research Fund Research Excellence Program, and the Natural Sciences and
Engineering Research Council (NSERC) of Canada. Computations were performed on the
General Purpose Cluster supercomputer at the SciNet HPC Consortium. SciNet is funded by: the
Canada Foundation for Innovation under the auspices of Compute Canada; the Government of
Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. The
authors thank O. Voznyy and A. Jain for useful discussions.
SUPPORTING INFORMATION FOR PUBLICATION
Supporting Information. Imaginary components of the SHG susceptibility, structural
parameters following relaxation, lithium niobate benchmark, Born effective charges, atomic
eigendisplacements, phonon frequencies, mode oscillator strengths, mode effective charges, and
modal Raman susceptibilities. (PDF).
23
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